Some lessons seem to spontaneously skitter off-track into a mud puddle of confusion. It feels like the students are distracted, annoyed, and politely bored all at the same time. “Just be quiet and let us get our backpacks and start cleaning up.” Today is one of those. It’s hard to tell whether they’re mad at me because I’m frustrated with them or the other way around. Sometimes I try stopping instantly and asking them what they have planned for the weekend. (It takes a minute for them to warm to the topic and then they won’t shut up.)
I’ve decided that a couple of the many causes of this kind of confusion include (1) they don’t see where we’re going or the context of where we are, and (2) they hate algebra and don’t know “when they’re ever going to use this stuff.” Today, I try to head off the second problem by acknowledging that most people truly won’t do algebra every day for the rest of their lives, but we hope they get other things out of this class, such as the ability to think critically — “how do you know that?” for example. I even indulge in a little harmless indoctrination: “For example, how you know there are weapons of mass destruction? How do you know sending 21,000 more troops will fix everything and everybody will get to come home?”
My solution to the second problem is to post a small whiteboard next to my main board with Today’s Questions on it. These questions are today’s objectives, turned around: How can I write an equation in slope-intercept form, given a slope and the y-intercept? What is slope-intercept form? I tell the students that each day they should keep an eye on these questions, and by the end of the lesson, they should know how to answer them.
Today’s lesson was humbling. The lesson itself was about the slope-intercept form of a linear equation, and part of the small-group activity was to build a table of values. I demonstrated the process, gave the students specific instructions on what values to choose for x, displayed complete instructions on the overhead, and by third period figured out that nobody had a clue what I was talking about. They were confused about the example on the board, which I had clearly labeled “Example,” and half the class copied the x and the y values directly from it. We studied input-output tables a month ago, so I’m a little surprised that the students acted like they had never seen these in their lives.
I spent a lot of my childhood in many different books, including my father’s old college textbooks. One analytic geometry book in particular provided all kinds of activities; I made a lot of input-output tables in order to graph curves such as parabolas. This concept was already familiar when I had to do it in various math and science classes. It’s a real eye-opener to find dozens of kids who need to have this explained to them over and over.
As a result of this lesson, I added a mini-lesson on tables of values to my next lesson, and I’ll probably continue to do it. Our district tests have given low priority to functions and relations, so this has indirectly contributed to our skimming over tables. I’ll add some table building to my sub plans for my absence day (when I’ll be in San Diego for the math teacher conference).
I’ve stretched the slope-intercept lesson another day because of the class’s unfamiliarity with tables of values. (They were meant to build a five-row table of values and then plot the points in order to draw a line.)
In addition to the problems understanding the use and purpose of a table of values, the students are still having troubles with slope. A few of the more attentive students have seen the pattern where slope emerges from five points in the table: I’ve required the students to work with the x values of -2, -1, 0, 1, and 2, and this means the differences between subsequent vales in the y side of the table are each the slope of the equation. It makes me wonder if (when?) I should do a mini-lesson on these things. Clearly, I need to incorporate these tables in lessons at every opportunity.
Another teacher recommends adding a third column to input/output tables, between the x and the y column, which would be the expression equivalent to y. So in the equation y = 3x + 4 there would be an x column, a y column, and in the middle a 3x + 4 column, where the substitution and simplifying happen. It bothers me a bit because it seems to duplicate effort, screw up the peer/peer relationship between x and y, and feels like coddling. I’ve resisted doing this for a couple lessons now. The trouble with not doing this is I can’t really tell the students where they should be doing the evaluation for each row; it’s kind of just hanging out there with no home. Happily, I think all my repetition of the steps of evaluating an expression is paying off: I say, “Next step? Next, after we write down the expression? I’m thinking of a word that starts with S?” And the majority of the responses are “substitute,” outnumbering “simplify” and almost completely wiping out “solve.”
I’m really trying to anticipate and forestall this “I don’t really get this stuff” lazy-ass blanket response. Half the time I hear this, it’s because someone hasn’t been paying attention. Still my problem. The other half the time, does that mean I haven’t been doing my job right? I’m trying to give them context, motivation, a sense of esprit de corps. I present our daily destination three different ways these days: questions on the Today’s Questions board, as goals (described on the main board in the housekeeping section and in the slide presentation), and as graphical shorthand (a graph symbol, an arrow, an equation – representing that we can derive an equation from a graph). Still they say, “I don’t really get any of this stuff.” I need to encourage the outgoing 10% to paraphrase our destination for their peers.
I gave a few token writing assignments and an easy quiz for the first week of the semester, so that a maximum number of students start off on the right foot. Half of the students now have honest-to-god A’s; for the rest, I’m offering makeup assignments. One of my troublemakers actually saw “A” next to his student ID and had to ask me to confirm what he was seeing, apparently a rare event in his life. (Imagine a cartoon character rubbing his eyes and blinking in disbelief.)
Today’s lesson in conversion of point-slope form to slope-intercept went pretty well, other than the predictable roadblocks: fractions and integer arithmetic. They brought a fear of fractions with them from elementary school. I’ve tried mocking it, lightheartedly: “I know you all got up this morning, and the first thing you thought was, ‘Dude. I hate fractions.’” And I’ve tried “counseling”: “It’s going to be okay. We’re not even doing anything with this fraction. We’re just writing it down. It’s not going to hurt. … There. Now, did that hurt?” This is an ancient dilemma, and I know a lot of smart-ass comments, but I don’t know a solution. I will leave them with a fraction-simplification worksheet while I’m away Friday, and I will prep them for this Thursday. Maybe that will help.
And then there’s the lingering inability to work with negative signs during basic operations. I’ve tried reasoning with them – “if two times two is four, then would negative two be the same thing?” – but they don’t have enough number sense to be persuaded by such logic. I probably need to add more justification to (what seems to me to be the “paperwork”) part of the exercises: simplification. The student should be able to explain why -4 + -2 = 6, rather than just hearing me tell them it’s wrong.
Continuing discussion of point-slope form and conversion. Apparently, some students think that when you subtract the same amount from each side of an equation, that’s just for the assignments where the book says “solve” or “isolate”; we’re now seeing students who can’t make the next step when all it takes to get to y = mx+b is to subtract 2 from each side. On the other hand, I seem to be seeing fewer messed up distributions than a couple months ago.
Students continue to have trouble reading a graph. The book presents a line with a clearly marked point (labeled) and arrows representing rise and run. About half the class can identify the point, fewer can identify the rise and run, and fewer still get the slope from this. I need to develop a mini-lesson on identifying these elements of the line of a graph.